IN THIS NOTE, WE PROVE THE EXISTENCE AND UNIQUENESS OF WEAK SOLUTIONS FOR THE BOUNDARY VALUE PROBLEM MODELLING THE STATIONARY CASE OF THE BIOCONVECTIVE FLOW PROBLEM. THE BIOCONVECTIVE MODEL IS A BOUNDARY VALUE PROBLEM FOR A SYSTEM OF FOUR EQUATIONS: THE NONLINEAR STOKES EQUATION, THE INCOMPRESSIBILITY EQUATION, AND TWO TRANSPORT EQUATIONS. THE UNKNOWNS OF THE MODEL ARE THE VELOCITY OF THE FLUID, THE PRESSURE OF THE FLUID, THE LOCAL CONCENTRATION OF MICROORGANISMS, AND THE OXYGEN CONCENTRATION. WE DERIVE SOME APPROPRIATE A PRIORI ESTIMATES FOR THE WEAK SOLUTION, WHICH IMPLIES THE EXISTENCE, BY APPLICATION OF GOSSEZ THEOREM, AND THE UNIQUENESS BY STANDARD METHODOLOGY OF COMPARISON OF TWO ARBITRARY SOLUTIONS.

THIS PAPER DEALS WITH A STUDY ON AN INITIAL-BOUNDARY VALUE PROBLEM FOR INCOMPRESSIBLE NON-NEWTONIAN FLUIDS OF DEGREE TWO, IN A BOUNDED AND SIMPLY CONNECTED OPEN SET ? OF R3. IT WILL BE SHOWN THAT THE GLOBAL-IN-TIME WEAK SOLUTION IS UNIFORMLY STABLE (I.E. THE BEHAVIOR OF THE SOLUTION CHANGES CONTINUOUSLY WITH THE DATA), AND CONSEQUENTLY THIS SOLUTION IS UNIQUE. FURTHERMORE, AN EXPONENTIAL DECAY RATE FOR THE ENERGY OF THE WEAK SOLUTION WILL ALSO BE ESTABLISHED.

IN THIS PAPER WE PROVE THE WELL-POSEDNESS AND WE STUDY THE ASYMPTOTIC BEHAVIOR OF NONOSCILLATORY LP-SOLUTIONS FOR A THIRD ORDER NONLINEAR SCALAR DIFFERENTIAL EQUATION. THE EQUATION CONSISTS OF TWO PARTS: A LINEAR THIRD ORDER WITH CONSTANT COEFFICIENTS PART AND A NONLINEAR PART REPRESENTED BY A POLYNOMIAL OF FOURTH ORDER IN THREE VARIABLES WITH VARIABLE COEFFICIENTS. THE RESULTS ARE OBTAINED ASSUMING THREE HYPOTHESES: (1) THE CHARACTERISTIC POLYNOMIAL ASSOCIATED WITH THE LINEAR PART HAS SIMPLE AND REAL ROOTS, (2) THE COEFFICIENTS OF THE POLYNOMIAL SATISFY ASYMPTOTIC INTEGRAL SMALLNESS CONDITIONS, AND (3) THE POLYNOMIAL COEFFICIENTS ARE IN LP([T0, ?[). THESE RESULTS ARE APPLIED TO STUDY A FOURTH ORDER LINEAR DIFFERENTIAL EQUATION OF POINCARÉ TYPE AND A FOURTH ORDER LINEAR DIFFERENTIAL EQUATION WITH UNBOUNDED COEFFICIENTS. MOREOVER, WE GIVE SOME EXAMPLES WHERE THE CLASSICAL THEOREMS CANNOT BE APPLIED.

WE STUDY THE HOMOGENIZATION AND LOCALIZATION OF HIGH-FREQUENCY WAVES IN A LOCALLY PERIODIC MEDIA WITH PERIOD ?. WE CONSIDER INITIAL DATA THAT ARE LOCALIZED BLOCH-WAVE PACKETS, I.E. THAT ARE THE PRODUCT OF A FAST OSCILLATING BLOCH WAVE AT A GIVEN FREQUENCY ? AND OF A SMOOTH ENVELOPE FUNCTION WHOSE SUPPORT IS CONCENTRATED AT A POINT X WITH LENGTH SCALE . WE ASSUME THAT (?, X) IS A STATIONARY POINT IN THE PHASE SPACE OF THE HAMILTONIAN ?(?, X), I.E. OF THE CORRESPONDING BLOCH EIGENVALUE. UPON RESCALING AT SIZE WE PROVE THAT THE SOLUTION OF THE WAVE EQUATION IS APPROXIMATELY THE SUM OF TWO TERMS WITH OPPOSITE PHASES WHICH ARE THE PRODUCT OF THE OSCILLATING BLOCH WAVE AND OF TWO LIMIT ENVELOPE FUNCTIONS WHICH ARE THE SOLUTION OF TWO SCHRÖDINGER TYPE EQUATIONS WITH QUADRATIC POTENTIAL. FURTHERMORE, IF THE FULL HESSIAN OF THE HAMILTONIAN ?(?, X) IS POSITIVE DEFINITE, THEN LOCALIZATION TAKES PLACE IN THE SENSE THAT THE SPECTRUM OF EACH HOMOGENIZED SCHRÖDINGER EQUATION IS MADE OF A COUNTABLE SEQUENCE OF FINITE MULTIPLICITY EIGENVALUES WITH EXPONENTIALLY DECAYING EIGENFUNCTIONS.

IN THIS PAPER, WE INTRODUCE THE FUNCTIONAL FRAMEWORK AND THE NECESSARY CONDITIONS FOR THE WELL-POSEDNESS OF AN INVERSE PROBLEM ARISING FROM THE MATHEMATICAL MODELING OF DISEASE TRANSMISSION. THE DIRECT PROBLEM IS GIVEN BY AN INITIAL BOUNDARY VALUE PROBLEM FOR A REACTION-DIFFUSION SYSTEM. THE INVERSE PROBLEM CONSISTS IN THE DETERMINATION OF THE DISEASE AND RECOVERY TRANSMISSION RATES FROM OBSERVED MEASUREMENT OF THE DIRECT PROBLEM SOLUTION AT THE FINAL TIME. THE UNKNOWNS OF THE INVERSE PROBLEM ARE THE COEFFICIENTS OF THE REACTION TERM. WE FORMULATE THE INVERSE PROBLEM AS AN OPTIMIZATION PROBLEM FOR AN APPROPRIATE COST FUNCTIONAL. THEN, THE EXISTENCE OF SOLUTIONS OF THE INVERSE PROBLEM IS DEDUCED BY PROVING THE EXISTENCE OF A MINIMIZER FOR THE COST FUNCTIONAL. MOREOVER, WE ESTABLISH THE UNIQUENESS UP AN ADDITIVE CONSTANT OF THE IDENTIFICATION PROBLEM. THE UNIQUENESS IS A CONSEQUENCE OF THE FIRST ORDER NECESSARY OPTIMALITY CONDITION AND A STABILITY OF THE INVERSE PROBLEM UNKNOWNS WITH RESPECT TO THE OBSERVATIONS.

ABSTRACT WE TREAT THE EXISTENCE AND UNIQUENESS OF REPRODUCTIVE SOLUTION (WEAK TIME-PERIODIC SOLUTION) OF A SECOND-GRADE FLUID SYSTEM FOR SMALL ENOUGH SOURCE TERMS, BY USING THE GALERKIN APPROXIMATION METHOD AND COMPACTNESS ARGUMENTS. RÉSUMÉ ON TRAITE LEXISTENCE ET LUNICITÉ DE LA SOLUTION REPRODUCTIVE DUN SYSTÈME DE FLUIDE DE GRADE DEUX AVEC DES TERMES SOURCES SUFFISAMMENTS PETITS, EN UTILISANT LA MÉTHODE DAPPROXIMATION DE GALERKIN ET DES ARGUMENTS DE COMPACITÉ.

WE CONSIDER THE EXISTENCE AND UNIQUENESS OF PERIODIC SOLUTIONS FOR THE GENERALIZED BIOCONVECTIVE FLOW, WHICH IS A WELL-KNOWN MODEL TO DESCRIBE THE CONVECTION CAUSED BY THE CONCENTRATION OF UPWARD SWIMMING MICROORGANISM IN A FLUID.